If you search the Web for the history of fitted sheets, you’re going to find a reference to exactly two patents:
1959 - Bertha Berman of New York: FITTED BED SHEET CONSTRUCTION (PDF, 355Kb)
1992 - Gisele Jubinville of Alberta: MATTRESS COVER/FITTED BED SHEET (PDF, 492Kb)
Now, if you actually look at the patents, what you’ll see is that Berman’s was a more complex design that consisted of multiple pieces (but does feature elastic!) And Jubinville’s design, indeed, has the stitched-in elastic design of the mattress-hugging fitted sheet creature we all love to hate today.
But I was alive before 1992 and I can assure you that we had fitted sheets when I was a kid and they weren’t anything like the Berman design.
It doesn’t add up.
A patent search reveals a number of intermediate designs. Some are wacky. But some look very much like a modern fitted sheet. Take this one, for example:
1965 - Belvin Roddey of South Carolina: FITTED BED SHEET AND METHOD FOR MAKING SAME (PDF, 633Kb)
I’m not saying Belvin Roddey’s design is the design used for decades before Jubinville’s design, I’m just saying there were definitely some steps between 1959 and 1992. (To confirm this, you can find fitted sheets from the '80s on Ebay. I saw a set with Disney’s Snow White that looked like she was staring across the decades into an abyss. Sure enough: Elastic at all four corners, just like I remember from my glow-in-the-dark robot space battle set.)
For additional historical tidbits, this article has some fun facts and vintage pictures: Fitted sheets were a fab 50s fad that never faded (clickamericana.com)
A 1954 article about the "popular new fitted sheets"!
Fitted sheets (not elastic) were originally invented in the 1940s for baby cribs.
A set of instructions for how-to-fold a fitted sheet (pre-elastic!)
The point is that fitted sheets have had an 80 year history that pre-dates the use of elastic.